With some work, this could be a good 3/8 pilot.  This is for Hall-Toledo seat grinders (like the model EJ or VIP)

• Included is a threaded rod which is too long, you'll have to shorten it and run a die over it to cut some new threads.
• Tightening nut is ok
• Correct 3/8 expanding collar
• INCORRECT round end cap, this measures .376, and is probably for a 7/16 or maybe 10mm pilot.  But it does fit over the end of the pilot, and if you turned the OD down with some sandpaper in a drill to get it down to .360 or .365, it would probably work great.

3/8 x 7/16 top.

Actual top measurement is (measured in several places): .4350" - .4362"

Overall length is 7 -7/16"

R10